There are two possibilities:
1. The grease in the freewheel has congealed and thickened. When this happens, the pawls inside get gummed up and don't spring out to catch the ratchet quickly enough when you go from coasting to pedaling. Eventually, they pop out and catch. You said that pedaling steadily prevented it from slipping, so this is the most likely problem.
Here's what you do: take the rear wheel off the bike and put it in your lap with the freewheel facing up. Give the freewheel a spin by hand. You should be able to see the small space in between the spinning and the stationary part of the freewheel. Take some kind of light oil - Tri-Flow or other thin chain lubes are great for this - and apply it in this space while spinning the freewheel back by hand. This will allow it to get into the body of the freewheel and work its magic. If you still have trouble, try doing this treatment with WD-40, waiting a while, then using light oil again. This should solve your problem.
2. It's possible, but unlikely, that either the pawls or pawl springs in your freewheel are busted. If this is the case, you need a new freewheel. I doubt that this is the case. If you can't solve your problem with the steps described above, you might as well just get a new freewheel - they're cheap, and they can be had for free if you salvage from another 10-speed.
Good luck!